A rope of uniform mass m and length L is hanging from a support. A block of mass M is attached to the rope at the lower end. If a transverse pulse is produced at the lower end, find the time taken by the pulse to reach the upper end.
A
2√Lg[√(1+Mm)−√Mm]
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B
32√Lg[√(2+Mm)−√Mm]
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C
√Lg[√(1+Mm)−√Mm]
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D
32√Lg[√(1+Mm)−√Mm]
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Solution
The correct option is A2√Lg[√(1+Mm)−√Mm]
Linear mass density of rope (μ)=mL Tension at point A, TA=μxg+Mg Velocity of transverse pulse at point A v=√TAμ=√(μx+M)gμ ⇒dxdt=√(x+Mμ)√g[∵v=dxdt] To find time taken for wave pulse to travel to the upper end, integrate both sides setting the limit of x from 0 to L ⇒∫L0dx√(x+Mμ)=√g∫tt=0dt ⇒2[√(x+Mμ)]L0=√g[t]t0 ⇒2[√(L+Mμ)−√Mμ]=√gt ⇒t=2[√(L+Mμ)−√Mμ]√g=2√L+MLm−2√MLm√g[∵μ=ML] ⇒t=2√Lg[√(1+Mm)−√Mm]